Karlson's final "Acoustic Transducers" patent 3540544 said an elliptical reflector could improve evenness of response and dispersion.
There are FOUR Karlson X15 at Ebay with their 3-panel ellipse approximation and slotted pipe tweeter.
https://www.ebay.com/itm/334448068543?hash=item4ddea887bf:g:k7YAAOSwdOdih~5r
X15 was introduced in the fall of 1965 and originally sported a mini - clam/klam tweeter with 3" cone speaker to augment their CTS 15 inch woofer.
Karlson expert Carl Neuser favors curved reflectors in his builds. I'm pretty sure his X15 would be to regular size (28x20x15) so the reflector can not be vastly different than JK's X15 and still fit. Can simple ray tracing determine what works well? My t41" tall K18 sounds great - I think its height could be reduced to 36".
Its too bad there's a lack of appreciation for the Karlson design as something like Akabak 3 an a lot of modeling might come of with some answers without physical prototype.